Factorization for hardy spaces and characterization for BMO spaces via commutators in the bessel setting

Fix λ > 0. Consider the Hardy space H¹(R₊, dm_λ) in the sense of Coifman and Weiss, where R₊ := (0,∞) and dm_λ := x^2λ dx with dx the Lebesgue measure. Also, consider the Bessel operators

(equations present)


on R₊. The Hardy spaces H¹_Δλ and H¹_Sλ associated with Δ_λ and S_λ are defined via the Riesz transforms R_Δλ := ∂x(Δ_λ)^−1/2 and R_Sλ := x^λ ∂_xx^−λ(S_λ)^−1/2, respectively. It is known that H¹_{Δ λ} and H¹(R₊, dm_λ) coincide but that they are different from H¹_Sλ . In this article, we prove the following:
(a) a weak factorization of H¹(R₊, dm_λ) by using a bilinear form of the Riesz transform R_Δλ , which implies the characterization of the BMO space associated with Δ_λ via the commutators related to RΔ_λ ;
(b) that the BMO space associated with S_λ cannot be characterized by commutators elated to R_Sλ , which implies that H¹_Sλ does not have a weak factorization via a bilinear form of the Riesz transform R_Sλ.